The Straight Blow Up Section Family.

The straight blow up section family of a projection $Xtimes Yto X$ along a closed subscheme $Z$ of $Xtimes Y$ is an object that combines the universal properties of the universal section family of $Xtimes Yto X$ and of the blow up of $Xtimes Y$ along $Z$. It is our attempt to characterise a unique morphism between universal schemes of relatives clusters who emerged in the authoru0027s previous work. To construct the straight blow up section family first we develop a scheme theoretic version of the Chevalleyu0027s theorem for projections and for the codimension zero stratum.